Lecture 18

Final Project Design your own survey! Find an interesting question and population Design your sampling plan Collect Data Analyze using R Write 5 page paper on your results Due December 1

Final presentation All are required to give a short presentation Last two classes (December 1 and 6) Select one of the three projects Make a powerpoint presentation (no more than 3-5 slides) Present your results to the class

Statistics Main ideas of statistics Given multiple plausible models select one (or several) that is (are) the most consistent with the observed data Quantify a measure of belief in our solution The main idea is that if something looks like a very unlikely coincidence we would prefer another more likely explanation

Parameters and statistics Example: A random sample of 603 voters are asked if they prefer cats or dogs. http://news.nationalgeographic.com/news/2013/06/130619-pets-poll-animals-united-states-nation-dogs-cats/ 314 (52%) prefer dogs The observed percentage in the sample, 52%, is a statistic. The unknown percent of the population who would say yes is a parameter. Presumably it’s close to 52%, but we will never know. Question we’d like to ask: “How likely is the statistic to be how close to the parameter?”

Experiment Population (urn) – our class (x people) Question: Dogs vs Cats We will sample 5 Assign numbers Use R to sample Repeat 5 times ascertain truth and run R

US Polls 2016 estimate of the number of eligible voters is 225,778,000. We sampled 603 people at random and got 314 dogs.

Main idea We learned that given a model we can check if the data is consistent with it Idea: Find models that are consistent with the data.

US Polls 2016 estimate of the number of eligible voters is 225,778,000. We sampled 603 people at random and got 314 dogs. We will consider various models: True proportion is p = 0.001, 0.002, 0.003, … Which of the models is the data in agreement with?

Results

“P-value” Consider proportion of fake samples < 314 Values close to 0 or 1 are not consistent with the model. (Why?)

Cutoff Using better resolution of models A usual cutoff 0.05 split between both sides (0.975 and .025) Models selected: [.482, .560]

Fake data 3 sub-groups The sample contains: 46,392,000 (R) Sampled 111, 101 yes (R) Sampled 166, 30 yes (D) Sampled 326, 183 yes (I)

Issue Too many levers to “fiddle” (three different Ks for each group) Not easy to simply look how well the data fits for all models. Needs more sophisticated statistics

Naïve parametric bootstrap Compute a number/numbers estimated from the data (point estimate). Use this number to simulate a lot of fake data and see how the fake data can vary. Use this variability to estimate the uncertainty in our estimator

Without Stratification Estimated K= 225,778,000 * (314/603)= 117,569,307 Estimated p in (.481,.561)

Stratified problem Estimate K for each group – combine to get joint p estimate pcombined=.20*.91 +.29*.18 +.51*.56=.50 This is small difference by using stratification There is a (small) gain in uncertainty Bootstrap interval (.472,.535)

Bootstrap Stratified Blah

Bootstrap SRS Blah

Homework Modify the bootstrap codes to get the bootstrap intervals for the population proportion of yes in the following two situations: Population of 30000; SRS sample of 100, observe 32 yes. Population of 30000 stratified to subpopulations of 10000 and 20000. Sample 30 observing 27 from fist subpopulation and sample 70 observing 5 from second subpopulation.